Chapter 6
Conclusion and future work
This chapter presents a general overview of the work done in this thesis. The aim of thesis was to develop efficient RBDO methods that can generate better reliable solutions as compared to the existing algorithms in the literature and can reduce the computational cost of these methods. Both numerical-based methods and meta-heuristic algorithms have been developed for solving single-objective and multi-objective RBDO examples. Uncertainties in the form of random variables were treated with low probability of failure of constraints to obtain reliable solution. All the developed methods were tested on several mathematical and engineering benchmark examples and compared with existing RBDO methods to validate the results.
solutions. The method was tested with eight benchmark examples from the literature.
From the results it can be concluded that SLShV-CG converged to the desired reliable solution irrespective of the nature of the performance function with better computational efficiency compared to the chosen set of RBDO methods. However, from example 2 it can be observed that SLSV-CG encounters with an issue of oscillation during convergence.
As SLShV-CG method was developed using conjugate gradient search direction for ap- proximating MPTP, it may have a tendency to oscillate while converging to the reliable solution for highly non-linear performance functions. This leads to the second objective of the thesis.
For addressing this issue ASLCC-2 was proposed in the second part of chapter 3. The oscillation criterion was proposed to estimate the chaos among consecutive MPTPs dur- ing convergence in the standard normal space. In every iteration, MPTP was estimated using conjugate gradient search directions and the last three MPTPs were used to track the oscillation among them. When oscillation was observed, chaos control theory was used to update the search direction of current MPTP. An oscillation criterion was also proposed to track these oscillations in the standard normal variable space. By solving the mathematical and engineering RBDO examples it can be observed that ASLCC-2 successfully eliminated the issue of oscillation while convergence. Example 2 also demon- strated smooth convergence when solved by ALSCC-2. Although, ALSCC-2 successfully mitigates the issue of oscillation, it has a tendency to converge to a local solutions. This leads to the motivation of third objective that is to obtain a reliable global solution.
• In the literature, there exists a single-loop method which was able to generate global solution. It was developed by using modified roulette wheel selection with stochastic acceptance in order to sort out solutions for mutation. Therefore, there was a need to explore the capabilities of other mutation schemes for obtaining a global reliable solution.
In chapter 4, a single-loop reliability-based design optimization with adaptive DE has been proposed to address the issue of global convergence. A single-loop formulation was presented with shifting vector approach incorporating target and trial vectors. DE was made adaptive using the heuristic parameter that helped DE to perform different mutation operators. The proposed method, SLADE, was tested on three mathematical and four engineering RBDO examples. The results demonstrated that the proposed method successfully converged to the reliable global optimal solution for the chosen set of examples. The proposed method was also found computationally efficient than the
DLRBDO-DE method. Although, SLADE was able to generate reliable global solution, it was limited to single objective examples. It has been observed that most of the real- world problems consist of multiple objectives apart from uncertainty. This leads to the development of fourth objective that is an single-loop multi-objective reliability-based design optimization method.
• In the literature, many meta-heuristic algorithms were developed to obtain reliable PO solutions for multi-objective RBDO problems. All these methods were based on the plat- form of either double-loop or decoupled-loop method, which makes it computationally expensive. Therefore, efficient single-loop method should be developed to solve those problems. In chapter 5, for obtaining a set of reliable PO solutions a single-loop multi- objective formulation has been proposed incorporating chaos control theory for estimating the most probable target point. The formulation was solved using differential evolution with adaptive mutation scheme. A heuristic convergence parameter was proposed to perform different mutation schemes for better exploration and exploitation in the search space. The proposed SL-MODE was tested on two mathematical and one engineering bi-objective RBDO examples. It can be concluded from the results that the proposed SL-MODE generated the better reliable Pareto-optimal solutions than DL-MODE. Also, the convergence of SL-MODE observed less fluctuations than DL-MODE due to the estimation of MPTP using chaos control theory. Moreover, SL-MODE was found com- putationally efficient than DL-MODE by requiring less number of function evaluations.
It has been observed in chapter 3 that both shifting vector approach and chaos control theory perform well solving RBDO examples. Therefore, in the last objective of this thesis both the concepts are incorporated for multi-objective RBDO formulation.
In the last objective, a single-loop multi-objective reliability-based design optimization has been proposed with shifting vector approach for generating reliable PO solutions. The search direction of approximate MPTP was modified by chaos control theory. The con- cept of shifting vector approach was implemented with the novel multi-objective RBDO formulation to include both target and trial vectors. The proposed SLMDE was tested on the same mathematical and engineering bi-objective RBDO examples. It was found that SLMDE generated better reliable PO solutions for all examples compared to DLMDE.
From the above discussion, it can be concluded that this thesis is focused on development of single-loop reliability-based design optimization methods for solving single and multi-objective
optimization problem. Various issues related to single-loop method have been addressed in this study and the scope of future work is discussed in the next section.